116 lines
3.2 KiB
Typst
116 lines
3.2 KiB
Typst
#import "@local/handout:0.1.0": *
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#import "@preview/cetz:0.3.1"
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= Rotational Symmetry
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Another symmetry is n-fold rotational symmetry about a point, whose signature is written n. Multiple bold numbers means multiple points of rotational symmetry.
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Two points of rotational symmetry are considered the same if we can perform a translation + rotation sending one to the other, while leaving the pattern the same.
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There are also patterns with both kinds of symmetries. To classify such patterns, first find all the mirror symmetries, then all the rotational symmetries that are not accounted
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for by the mirror symmetries.
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By convention we write the rotational symmetries before
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the `*`.
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#table(
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stroke: none,
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align: center,
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columns: (1fr, 1fr),
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rows: 50mm,
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image("../res/333.png", height: 100%), image("../res/3*3.png", height: 100%),
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)
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#problem()
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Mark the three rotation points in Figure 1.
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#problem()
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Find the signature of the pattern in Figure 2.
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#solution([`3 *3`])
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#pagebreak()
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Some exceptional cases: It is possible to have two different parallel mirror lines. In
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this situation the signature is ∗ ∗
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#table(
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stroke: none,
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align: center,
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columns: 1fr,
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rows: 60mm,
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image("../res/**.png", height: 100%),
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)
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#problem()
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Draw another wallpaper pattern with signature `**`
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#pagebreak()
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There are two other types of symmetries. The first called a miracle whose signature is
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written ×. It is the result of a glide reflection, which is translation along a line followed
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by reflection about that line.
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This occurs when there is orientation-reversing symmetry not accounted for by a mirror.
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For example, if we modify Figure 3 slightly we get a signature of ∗×
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#table(
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stroke: none,
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align: center,
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columns: (1fr, 1fr),
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rows: 60mm,
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image("../res/*x-b.png", height: 100%),
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image("../res/*x-a.png", height: 100%),
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)
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Signature ∗×. There is a glide reflection (shown by the by the dotted line)
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taking the clockwise spiral to the counter-clockwise spiral, reversing orientation
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#pagebreak()
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#problem()
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Find the signatures of the following patterns:
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#table(
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stroke: none,
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align: center,
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columns: (1fr, 1fr),
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rows: 60mm,
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image("../res/wiki/Wallpaper_group-cm-4.jpg", height: 100%),
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image("../res/wiki/Wallpaper_group-p4g-2.jpg", height: 100%),
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)
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#pagebreak()
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There is another exceptional case with two miracles, where there are two glide reflection
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symmetries along distinct lines. There are other glide reflections, but they can be obtained
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by composing the two marked in the diagram.
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#table(
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stroke: none,
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align: center,
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columns: (1fr, 1fr),
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rows: 60mm,
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image("../res/xx-b.png", height: 100%),
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image("../res/xx-a.png", height: 100%),
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)
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Figure 7: There are two distinct mirrorless crossings, so the signature is `xx`.
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Lastly, if none of the above symmetries appear in the pattern, then there is only regular
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translational symmetry, which we denote by O.
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In summary, to find the signature of a pattern:
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- Find the mirror lines (∗) and the distinct intersections
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- Find the rotational points of symmetry not account for by reflections.
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- Look for any miracles (×) i.e. glide reflections that do not cross a mirror line.
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- If you found none of the above, it is just O
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